1. CSC-2259 Discrete Structures Konstantin Busch Louisiana State University
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2. Topics to be covered Logic and Proofs Sets, Functions, Sequences, Sums
Integers, Matrices
Induction, Recursion
Counting
Discrete Probability
Graphs
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3. Binary Arithmetic Decimal Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Numbers: 9, 28, 211, etc
Binary Digits: 0, 1
(also known as bits)
Numbers: , 11100, , etc
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4. Binary
Decimal
1001 9
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5. 1001 (9) 1001 (9) x 1 1 (3) + 1 1 (3) ------ ------ 1001 1100 (12)
Binary Addition
Binary Multiplication
(9)
x (3)
------
1001
+ 1001
(27)
(9)
(3)
------
(12)
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6. Binary Logic AND OR NOT Gates NOT AND OR x y z 1 x y z 1 x z 11 x y z 1
NOT x z 1
Gates
NOT
AND OR
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7. … An arbitrary binary function is implemented
with NOT, AND, and OR gates … OR
AND
NOT
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8. Propositional Logic Proposition: a declarative sentence which
is either True or False
Examples:
Today is Wednesday (False)
Today it Snows (False)
1+1 = (True)
1+1 = (False)
H20 = water (True)
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9. We can map to binary values: True = 1 False = 0
Propositions can be combined using
the binary operators AND, OR, NOT
Example:
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10. “You get a computer science degree
Implication
True
False
x implies y
“You get a computer science degree
only if you are a computer science major”
You get a computer science degree
You are a computer science major
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11. “There is a received phone call if and only if there is a phone ring”
Bi-conditional
True
False
x if and only if y
“There is a received phone call if and only if
there is a phone ring”
There is a received phone call
There is a phone ring
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12. Sets Set is a collection of elements: Real numbers R Integers Z
Empty Set
Students in this room
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13. Basic Set Operations 1
Subset 3 2 4 5
Union 1 4 3 2 5
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14. Intersection 1 4 3 2 5
Complement
universe
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15. DeMorgan’s Laws
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16. Inclusion-Exclusion A B C
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17. Contains all subsets of a set
Powersets
Contains all subsets of a set
Powerset of A
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18. Counting Suppose we are given four objects: a, b, c, d
How many ways are there
to order the objects?
a,b,c,d
b,a,c,d
a,b,d,c
b,a,d,c … and so on
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19. Combinations Given a set S with n elements
how many subsets exist with m elements?
Example:
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20. Sterling’s Approximation
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21. Probabilities What is the probability the a dice gives 5?
Event set = {5}
Sample space = {1,2,3,4,5,6}
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22. What is the probability that two dice give the same number?
Event set = {{1,1},{2,2},{3,3},{4,4},{5,5},{6,6}}
Sample Space = {{1,1},{1,2},{1,3}, …., {6,5}, {6,6}}
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23. Quicksort(A): Randomized Algorithms If ( |A| == 1)
return the one item in A
Else
p = RandomElement(A)
L = elements less than p
H = elements higher than p
B = Quicksort(L)
C = Quicksort(H)
return(BC)
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24. Graph Theory San Francisco Chicago Boston New York Las Vegas Atlanta
1500 miles
Boston
2000 miles
300
New York
800
1500
1500
Las Vegas
800
1000
700
Atlanta
1000
1000
1500
2000
700
Baton Rouge
1500
Los Angeles
Miami
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25. Shortest Path from Los Angeles to Boston
1500
2000
300
800
1500
1500
800
1000
700
1000
1000
1500
700
2000
1500
Los Angeles
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26. Maximum number of edges in a graph with nodes:
Clique with five nodes
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27. Other interesting graphs
Trees
Bipartite Graph
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28. Recursion Sum of arithmetic sequence Basis Sum of geometric sequence
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29. Divide and conquer algorithms (Quicksort)
Fibonacci numbers
Basis
Divide and conquer algorithms (Quicksort)
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30. Proof Techniques Induction Contradiction Pigeonhole principle
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31. Proof by Induction Prove: Induction Basis: Induction Hypothesis:
Induction Step:
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32. Proof by Contradiction
is irrational
Suppose
( and have no common
divisor greater than 1 )
m=2k
m2 is even
m is even
2 n2 = 4k2
n2 = 2k2
n is even
Contradiction
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